Abstract/Summary

The plethora, and mass take up, of digital communication tech-
nologies has resulted in a wealth of interest in social network data
collection and analysis in recent years. Within many such networks
the interactions are transient: thus those networks evolve over time.
In this paper we introduce a class of models for such networks using
evolving graphs with memory dependent edges, which may appear and
disappear according to their recent history. We consider time discrete
and time continuous variants of the model. We consider the long
term asymptotic behaviour as a function of parameters controlling
the memory dependence. In particular we show that such networks
may continue evolving forever, or else may quench and become static
(containing immortal and/or extinct edges). This depends on the ex-
istence or otherwise of certain inﬁnite products and series involving
age dependent model parameters. To test these ideas we show how
model parameters may be calibrated based on limited samples of time
dependent data, and we apply these concepts to three real networks:
summary data on mobile phone use from a developing region; online
social-business network data from China; and disaggregated mobile
phone communications data from a reality mining experiment in the
US. In each case we show that there is evidence for memory dependent
dynamics, such as that embodied within the class of models proposed
here.